
where \(\sigma ^{\prime }\approx -0.0307\)  characterizes the second-order correction of the Fermi momentum, i.e., via \(k_{\mathrm {F}}\approx \overline{k}_{\mathrm {F}}(1+\sigma \epsilon +\sigma ^{\prime }\epsilon ^2)\) .
Tab. REF  gives the contributions from different models at order \(\epsilon ^2\)  to the symmetry energy.
In the ImMDI model, if the \(y\)  parameter is taken to be about \(-115\,\mathrm {MeV}\)  [1]}, then all the contributions to \(E_{\mathrm {sym}}(\rho _0)\)  at order \(\epsilon ^2\)  are negative.
Another feature is that for each model, the second-order contribution can not change the qualitative conclusion from the linear-order. For example, in the Skyrme model [2]} although the second-order contribution is negative the overall result is still positive determined by the linear term.
In this sense, the second-order term is really a “perturbation” (small correction) although \(\epsilon =-1\)  is applied for the 2D symmetry energy.
One can similarly define the effective form of the symmetry energy in 2D as in (REF ).
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